CN114964673B - Structural frequency response function correction method for frequency spectrum leakage error - Google Patents
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Abstract
The invention belongs to the field of modal identification of structural data analysis, and provides a structural frequency response function correction method for a frequency spectrum leakage error. And establishing a theoretical displacement frequency response function to be fitted by adopting a rational-fractal polynomial method, and performing curve fitting on the discrete actually-measured displacement frequency response data by adopting a least square method to finally obtain the fitted displacement frequency response function. And performing partial fractional expansion on the fitted displacement frequency response function, and then extracting an imaginary part of each order of modal residue to construct a modified structure displacement frequency response function. The method can directly eliminate the influence of the frequency spectrum leakage error on the structure displacement frequency response function, namely, the real displacement frequency response function of the structure is accurately obtained, the calculation process is clear and simple, and iteration is not needed.
Description
Technical Field
The invention belongs to the field of modal identification of structural data analysis, and relates to a frequency response function correction method of an engineering structure.
Background
The modal analysis method is one of important means for solving the dynamic characteristic design of the modern complex structure, and modal parameters reflect the dynamic characteristics of the structure and can be used for vibration control, model correction, structural state evaluation and the like, so that the modal parameter identification of the structure by using the measured data is very important.
The frequency response function reflects the transfer capability of the system to different input signals, and is a non-parametric model for describing the characteristics of the dynamic system. For any system, the stability of the system can be directly analyzed through a frequency response function, the system is comprehensively designed and corrected, a powerful tool is provided for application control theory analysis and solving of complex problems in engineering practice, and important positions are occupied in aspects of modal analysis, parameter identification, model correction, fault diagnosis and the like.
At present, frequency response function estimation methods are mature and have a plurality of types, and the commonly used frequency response estimation methods mainly comprise three methods, namely an H1 method, an H2 method and an Hv method. On this basis, the spectral leakage error is generally reduced by applying a windowing function to the time domain data, such as a rectangular window, a hamming window, an exponential window, and the like. Schoukens further compared the spectral leakage principle of rectangular windows and hamming windows, and indicated that hamming windows actually convert leakage errors of rectangular windows into interpolation errors. Antoni proposed a method of weighted overlap-average of data, demonstrating that leakage errors and random errors can be reduced by the overlap ratio of adjacent data blocks. When the modal parameters of the engineering structure are identified by actually measuring the frequency response function through a frequency domain method, the estimation precision of the frequency response function directly leads to the quality of the modal parameter identification effect. However, the above frequency response estimation method can only reduce the influence of the spectrum leakage error, thereby causing unstable modal parameter identification. Therefore, how to correct the structural frequency response function for the spectrum leakage error is crucial, which also has reference value for the practical application of many frequency domain identification methods.
Disclosure of Invention
The invention aims to provide a displacement frequency response function correction method for a proportional damping structure, and solves the problem that a fitted displacement frequency response function is inaccurate due to frequency spectrum leakage errors.
The technical scheme of the invention is as follows:
a structure frequency response function correction method aiming at a frequency spectrum leakage error comprises the steps of firstly respectively carrying out fast Fourier transform on a limited recording length displacement response and an excitation force signal of a structure under pulse excitation, and obtaining discrete measured displacement frequency response data through a frequency response calculation method. And establishing a theoretical displacement frequency response function to be fitted by adopting a rational fraction polynomial method, and performing curve fitting on the discrete actually-measured displacement frequency response data by adopting a least square method to finally obtain the fitted displacement frequency response function. Performing partial fractional expansion on the fitted displacement frequency response function, then extracting the imaginary part of each order of modal residue, calculating the corresponding spectrum leakage error, and finally constructing a corrected structure displacement frequency response function; the method comprises the following steps:
(1) Applying single-point pulse excitation to the structure, acquiring excitation force response by using a force sensor, acquiring displacement response at a measuring point of the structure by using a displacement sensor, and respectively transforming the excitation force response and the displacement response to a frequency domain by adopting fast Fourier transform to obtain a corresponding excitation force frequency spectrum and a corresponding displacement response frequency spectrum;
(2) Obtaining discrete actually-measured displacement frequency response data by utilizing the ratio of the displacement response frequency spectrum to the excitation force frequency spectrum at each frequency point;
(3) Adopting a rational numerator polynomial method to establish a theoretical displacement frequency response function to be fitted as follows:
wherein, N represents the number of peak values in the amplitude diagram of the actually measured displacement frequency response data; a. The k And B k Undetermined coefficients in numerator and denominator of a theoretical displacement frequency response function are respectively represented, k =0,1, \8230, 2N-1; j represents an imaginary unit and satisfies j 2 ＝-1；
(4) Performing curve fitting on actually measured displacement frequency response data by a least square method to obtain undetermined coefficient A k And B k Obtaining a fitted displacement frequency response function;
(5) And (3) carrying out partial fraction expansion on the fitted displacement frequency response function:
wherein, mu i Residue representing the ith mode, symbol "+" representing conjugate, s i A pole representing the ith order mode, i =1,2, \ 8230;, N;
(6) The frequency and damping ratio are found by:
w i ＝|s i |
wherein, w i And ζ i The ith order frequency and the damping ratio of the structure are respectively, | and Re (-) respectively represent the amplitude and the real part of the extracted data;
(7) The corrected residue is found by:
wherein T is the finite recording length of the displacement response,
damped frequencies for order i, imag (·) represents the imaginary part of the extracted data;
(8) Establishing a corrected structure displacement frequency response function:
the invention has the beneficial effects that: the method can directly eliminate the influence of the frequency spectrum leakage error on the structure displacement frequency response function, namely, the real displacement frequency response function of the structure is accurately obtained, the calculation process is clear and simple, and iteration is not needed.
Detailed Description
The following further illustrates embodiments of the present invention in conjunction with the technical solutions.
Taking a 3-layer frame structure as an example, the mass matrix and stiffness matrix are as follows:
the damping matrix adopts a Rayleigh damping matrix, namely C =0.2M. And applying pulse excitation to the layer 1 frame, wherein the response signal is the displacement of the layer 2 frame. The specific implementation mode of the method is as follows:
(1) Obtaining a finite record length displacement response x of a structure T (t)＝[x(t 1 ),x(t 2 ),…,x(t l )]And a pulsed excitation force signal f (t) = [ f (t) = 1 ),f(t 2 ),…,f(t l )]Respectively transforming the time domain displacement response and the force signal to a frequency domain by adopting fast Fourier transform, wherein the expression is X T (ω)＝[X T (ω 1 ),X T (ω 2 ),…,X T (ω l )]And F (ω) = [ F (ω) 1 ),F(ω 2 ),…,F(ω l )]Wherein l represents the number of data points of the displacement response or excitation force, ω i Represents the ith frequency point;
(2) Obtaining discrete measured displacement frequency response data by using the ratio of the displacement response frequency spectrum to the excitation force frequency spectrum, wherein the expression is
H T (ω)＝X T (ω)./F(ω)＝[H T (ω 1 )，H T (ω 2 )，…，H T (ω l )]
Wherein, the symbol "/" represents a dot division, that is, each column of elements in the vector is divided separately;
(3) By adopting a rational fraction polynomial method, a theoretical displacement frequency response function to be fitted is established as
Wherein, the number of peak values in the actually measured displacement frequency response data amplitude diagram expressed by N is 3 k (k =0,1, \ 8230;, 2N-1) and B k (k =0,1, \8230;, 2N) represents undetermined coefficients in numerator and denominator of theoretical shift frequency response function, respectively, j represents an imaginary unit and satisfies j 2 ＝-1；
(4) Performing curve fitting on actually measured frequency response data by a least square method to obtain undetermined coefficient A k And B k Obtaining a fitted displacement frequency response function;
(5) Partial fractional expansion is carried out on the fitted displacement frequency response function
Wherein, mu i Residue representing the ith mode of the frequency response function, symbol "+" representing the conjugate, s i A pole representing an ith order mode;
(6) The frequency and damping ratio are found by:
ω i ＝|s i |
wherein, ω is i And ζ i I-th order frequency and damping ratio of the structure, respectively, | and Re (·) represent the amplitude and real part of the extracted data, respectively.
(7) The corrected residue number [ mu' = j [ -5.3020, -1.7656, -2.3914 ] was obtained by the following equation]×10 -2 ：
Wherein T is the finite recording length of the displacement response,
damped frequencies for order i, imag (·) represents the imaginary part of the extracted data;
(8) Establishing a modified structure displacement frequency response function
Claims (1)
1. A structural frequency response function correction method for a frequency spectrum leakage error is characterized in that firstly, fast Fourier transform is respectively carried out on finite record length displacement response and excitation force response of a structure under pulse excitation, and discrete actual measurement displacement frequency response data are obtained through a frequency response calculation method; establishing a theoretical displacement frequency response function to be fitted by adopting a rational fraction polynomial method, and performing curve fitting on discrete actually-measured displacement frequency response data by adopting a least square method to finally obtain a fitted displacement frequency response function; performing partial fractional expansion on the fitted displacement frequency response function, then extracting the imaginary part of each order of modal residue and calculating the corresponding spectrum leakage error, and finally constructing a corrected structure displacement frequency response function; the method comprises the following steps:
(1) Applying single-point pulse excitation to the structure, acquiring excitation force response by using a force sensor, acquiring displacement response at a measuring point of the structure by using a displacement sensor, and respectively transforming the excitation force response and the displacement response to a frequency domain by adopting fast Fourier transform to obtain a corresponding excitation force frequency spectrum and a corresponding displacement response frequency spectrum;
(2) Obtaining discrete actually-measured displacement frequency response data by utilizing the ratio of the displacement response frequency spectrum to the excitation force frequency spectrum at each frequency point;
(3) Adopting a rational numerator polynomial method to establish a theoretical displacement frequency response function to be fitted as follows:
wherein, N represents the number of peak values in the amplitude diagram of the actually measured displacement frequency response data; a. The k And B k Undetermined coefficients in numerator and denominator of a theoretical displacement frequency response function are respectively represented, k =0,1, \8230, 2N-1; j represents an imaginary unit and satisfies j 2 ＝-1；
(4) Performing curve fitting on the actually measured displacement frequency response data by a least square method to obtain an undetermined coefficient A k And B k Obtaining a fitted displacement frequency response function;
(5) And (3) carrying out partial fractional expansion on the fitted displacement frequency response function:
wherein, mu i Residue representing the ith mode, symbol "+" representing conjugate, s i A pole representing the ith order mode, i =1,2, \ 8230;, N;
(6) The frequency and damping ratio are found by:
w i ＝|s i |
wherein, w i And ζ i The ith order frequency and the damping ratio of the structure are respectively, | and Re (-) respectively represent the amplitude and the real part of the extracted data;
(7) The corrected residue is found by the following equation:
wherein T is the finite recording length of the displacement response,
damped frequencies for order i, imag (·) representing the imaginary part of the extracted data;
(8) Establishing a modified structure displacement frequency response function:
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